Matrices can be added if they are the same order. The product of two matrices is defined if the number of columns of the first factor equals the rows of the second factor. A determinant provides a value for a square matrix and is calculated by summing terms with alternating signs based on the positions of elements. A system of linear equations is consistent if the determinant of the coefficient matrix is non-zero. Cramer's rule can be used to find the unique solution if it exists by taking the ratio of determinants involving the coefficient matrix and matrices with the constant column replaced.
Matrices can be added if they are the same order. The product of two matrices is defined if the number of columns of the first factor equals the rows of the second factor. A determinant provides a value for a square matrix and is calculated by summing terms with alternating signs based on the positions of elements. A system of linear equations is consistent if the determinant of the coefficient matrix is non-zero. Cramer's rule can be used to find the unique solution if it exists by taking the ratio of determinants involving the coefficient matrix and matrices with the constant column replaced.
Matrices can be added if they are the same order. The product of two matrices is defined if the number of columns of the first factor equals the rows of the second factor. A determinant provides a value for a square matrix and is calculated by summing terms with alternating signs based on the positions of elements. A system of linear equations is consistent if the determinant of the coefficient matrix is non-zero. Cramer's rule can be used to find the unique solution if it exists by taking the ratio of determinants involving the coefficient matrix and matrices with the constant column replaced.
Matrices can be added if they are the same order. The product of two matrices is defined if the number of columns of the first factor equals the rows of the second factor. A determinant provides a value for a square matrix and is calculated by summing terms with alternating signs based on the positions of elements. A system of linear equations is consistent if the determinant of the coefficient matrix is non-zero. Cramer's rule can be used to find the unique solution if it exists by taking the ratio of determinants involving the coefficient matrix and matrices with the constant column replaced.
Matrices : (iii) Nilpotent if Ak = O when k is a positive integer.
An m n matrix is a rectangular array of mn numbers Least value of k is called the index of the (real or complex) arranged in an ordered set of m nilpotent matrix. horizontal lines called rows and n vertical lines called (iv) Involutary if A2 = I. columns enclosed in parentheses. An m n matrix A The matrix obtained from a matrix A = [aij]m n by is usually written as : changing its rows into columns and columns of A into a11 a12 ... a1 j ... a1n rows is called the transpose of A and is denoted by A. a A square matrix a = [aij]n n is said to be 21 a22 ... a2 j ... a2 n M M (i) Symmetric if aij = aji for all i and j i.e. if A = A. A= (ii) Skew-symmetric if ai1 ai 2 ... aij ... ain M aij = aji for all i and j i.e., if A = A. M Every square matrix A can be uniquely written as sum am1 am 2 ... amj ... amn of a symmetric and a skew-symmetric matrix. Where 1 i m and 1 j n 1 1 1 A= (A + A) + (A A) where (A + A) is and is written in compact form as A = [aij]m n 2 2 2 A matrix A = [aij]m n is called 1 symmetric and (A A) is skew-symmetric. (i) a rectangular matrix if m n 2 (ii) a square matrix if m = n Let A = [aij]m n be a given matrix. Then the matrix (iii) a row matrix or row vector if m = 1 obtained from A by replacing all the elements by their (iv) a column matrix or column vector if n = 1 conjugate complex is called the conjugate of the matrix (v) a null matrix if aij = 0 for all i, j and is denoted by A and is denoted by A = [aij ] . O m n Properties : (vi) a diagonal matrix if aij = 0 for i j (vii) a scalar matrix if aij = 0 for i j and all diagonal ( ) (i) A = A elements aii are equal (ii) (A + B) = A + B Two matrices can be added only when thye are of same (iii) ( A) = A , where is a scalar order. If A = [aij]m n and B = [bij]m n, then sum of A and B is denoted by A + B and is a matrix [aij + bij]m n (iv) (A B) = A B . The product of two matrices A and B, written as AB, Determinant : is defined in this very order of matrices if number of Consider the set of linear equations a1x + b1y = 0 and columns of A (pre factor) is equal to the number of a2x + b2y = 0, where on eliminating x and y we get rows of B (post factor). If AB is defined , we say that the eliminant a1b2 a2b1 = 0; or symbolically, we A and B are conformable for multiplication in the write in the determinant notation order AB. a1 b1 If A = [aij]m n and B = [bij]n p, then their product AB a1b2 a2b1 = 0 is a matrix C = [cij]m p where a2 b2 Cij = sum of the products of elements of ith row of A Here the scalar a1b2 a2b1 is said to be the expansion with the corresponding elements of jth column of B. a b Types of matrices : of the 2 2 order determinant 1 1 having 2 a2 b2 (i) Idempotent if A2 = A rows and 2 columns. (ii) Periodic if Ak+1 = A for some positive integer k. Similarly, a determinant of 3 3 order can be The least value of k is called the period of A. expanded as :
XtraEdge for IIT-JEE 44 MAY 2011
a1 b1 c1 Let A be a square matrix of order n. Then the inverse of b2 c2 a2 c2 a2 b2 1 a2 b2 c 2 = a1 b1 + c1 A is given by A1 = adj. A. b3 c3 a3 c3 a3 b3 |A| a3 b3 c3 Reversal law : If A, B, C are invertible matrices of same = a1(b2c3 b3c2) b1(a2c3 a3c2) + c1(a2b3 a3b2) order, then = a1(b2c3 b3c2) a2(b1c3 b3c1) + a3(b1c2 b2c1) (i) (AB)1 = B1 A1 = ( aibjck) (ii) (ABC)1 = C1 B1 A1 To every square matrix A = [aij]m n is associated a number of function called the determinant of A and is Criterion of consistency of a system of linear equations denoted by | A | or det A. (i) The non-homogeneous system AX = B, B 0 has a11 a12 ... a1n unique solution if | A | 0 and the unique solution is given by X = A1B. a 21 a22 ... a2 n Thus, | A | = (ii) Cramers Rule : If | A | 0 and X = (x1, x2,..., xn) M M M | Ai | an1 an 2 ... a nn then for each i =1, 2, 3, , n ; xi = where |A| If A = [aij]n n, then the matrix obtained from A after Ai is the matrix obtained from A by replacing the deleting ith row and jth column is called a submatrix ith column with B. of A. The determinant of this submatrix is called a (iii) If | A | = 0 and (adj. A) B = O, then the system minor or aij. AX = B is consistent and has infinitely many Sum of products of elements of a row (or column) in solutions. a det with their corresponding cofactors is equal to (iv) If | A | = 0 and (adj. A) B O, then the system the value of the determinant. AX = B is inconsistent. n n (v) If | A | 0 then the homogeneous system AX = O i.e., i =1 aij Cij = | A | and a j =1 ij Cij = | A |. has only null solution or trivial solution (i.e., x1 = 0, x2 = 0, . xn = 0) (i) If all the elements of any two rows or two columns (vi) If | A | = 0, then the system AX = O has non-null of a determinant ate either identical or solution. proportional, then the determinant is zero. (i) Area of a triangle having vertices at (x1, y1), (x2, y2) (ii) If A is a square matrix of order n, then x1 y1 1 | kA | = kn | A |. 1 and (x3, y3) is given by x2 y 2 1 (iii) If is determinant of order n and is the 2 determinant obtained from by replacing the x3 y 3 1 elements by the corresponding cofactors, then (ii) Three points A(x1, y1), B(x2, y2) and C(x3, y3) are = n1 collinear iff area of ABC = 0. (iv) Determinant of a skew-symmetric matrix of odd A square matrix A is called an orthogonal matrix if order is always zero. AA = AA = I. The determinant of a square matrix can be evaluated A square matrix A is called unitary if AA = AA = I by expanding from any row or column. (i) The determinant of a unitary matrix is of modulus If A = [aij]n n is a square matrix and Cij is the unity. cofactor of aij in A, then the transpose of the matrix obtained from A after replacing each element by the (ii) If A is a unitary matrix then A, A , A, A1 are corresponding cofactor is called the adjoint of A and unitary. is denoted by adj. A. (iii) Product of two unitary matrices is unitary. Thus, adj. A = [Cij]. Differentiation of Determinants : Properties of adjoint of a square matrix Let A = | C1 C2 C3 | is a determinant then (i) If A is a square matrix of order n, then dA A . (adj. A) = (adj . A) A = | A | In. = | C1 C2 C3 | + | C1 C2 C3 | + | C1 C2 C3 | dx (ii) If | A | = 0, then A (adj. A) = (adj. A) A = O. Same process we have for row. (iii) | adj . A | = | A |n 1 if | A | 0 Thus, to differentiate a determinant, we differentiate one (iv) adj. (AB) = (adj. B) (adj. A). column (or row) at a time, keeping others unchanged. (v) adj. (adj. A) = | A |n 2 A.